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Realization of Lie algebras and classifying spaces of crossed modules

Yves Félix and Daniel Tanré

Algebraic & Geometric Topology 24 (2024) 141–157

The category of complete differential graded Lie algebras provides nice algebraic models for the rational homotopy types of nonsimply connected spaces. In particular, there is a realization functor, , of any complete differential graded Lie algebra as a simplicial set. In a previous article, we considered the particular case of a complete graded Lie algebra, L0, concentrated in degree 0 and proved that L0 is isomorphic to the usual bar construction on the Maltsev group associated to L0.

Here we consider the case of a complete differential graded Lie algebra, L = L0 L1, concentrated in degrees 0 and 1. We establish that the category of such two-stage Lie algebras is equivalent to explicit subcategories of crossed modules and Lie algebra crossed modules, extending the equivalence between pronilpotent Lie algebras and Maltsev groups. In particular, there is a crossed module 𝒞(L) associated to L. We prove that 𝒞(L) is isomorphic to the Whitehead crossed module associated to the simplicial pair (L,L0). Our main result is the identification of L with the classifying space of 𝒞(L).

rational homotopy, realization of Lie algebras, Lie models of simplicial sets, crossed modules
Mathematical Subject Classification
Primary: 17B55, 55P62
Secondary: 55U10
Received: 8 March 2021
Revised: 22 August 2022
Accepted: 3 September 2022
Published: 18 March 2024
Yves Félix
Département de Mathématiques
Université Catholique de Louvain
Daniel Tanré
Département de Mathématiques
Université de Lille

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